7/27/2019 Problem Set 17 Solutions
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PROBLEM SET 17 SOLUTIONS.
(1) Consider the matrixA =
1 0
1 2
.
(a) Find the eigenvalues ofA.(b) Find the eigenvectors ofA.(c)
Diagonalize A: write it as A = PDP1.
ANSWER:
(a) We just have to solve the characteristic equation
det
1 01 2
= (1 )(2 ) = 0.
So = 1, 2. Note that in general the eigenvalues of an upper
triangular or lowertriangular matrix are the diagonal entries.
(b) To find the eigenvector corresponding to , we have to find
the null space ofA I. There are two cases.
= 1. Here we have to solve0 0
1 1
a
b
=
00
So the eigenvector is 11
= 2. Here we have to solve1 01 0
a
b
=
00
So the eigenvector is 01
(2) Consider the matrix
A =
3 4 43 5 3
1 2 0
.
(a) Find the eigenvalues ofA. Just kidding! The eigenvalues are
1, 1 and 2. Twoeigenvectors are
v =
21
0
& w =
101
.
Check that these are eigenvectors. What are the corresponding
eigenvalues?1
7/27/2019 Problem Set 17 Solutions
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2 PROBLEM SET 17 SOLUTIONS
ANSWER: 3 4 43 5 31 2 0
21
0
=
21
0
So the eigenvalue corresponding to v is 1. 3 4 43 5 31 2 0
101
=
101
So the eigenvalue corresponding to w is 1.(b) Find a third
eigenvector corresponding to the third eigenvalue.
ANSWER: We need to find an element of the null space ofA 2I, in
other wordsa solution to
5 4 43 3 31 2 2
a
b
c
=
000
We can solve this using row reduction, or just observe that the
third column is just 1times the second column, and so from the
column definition of matrix multiplicationwe know immediately that
the eigenvector must be,
011
(Note: if you know about vector calculus, another neat way to
find the eigenvectoris to take the cross-product of any two rows
ofA I.)
(3) Suppose that A is an nn matrix, and that A2 = A. What can
you say, then, aboutthe eigenvalues ofA?
ANSWER: If is an eigenvalue ofA. Then
Av = v
Where v is an eigenvector corresponding to . Therefore
A2v = AAv = Av = Av = 2v
And so A2 = A implies A2v = Av . So it must be that
= 2
And must be 0 or 1.(4) Suppose A is a 3 3 matrix with
eigenvalues 1, 2 and 3. Ifv1 is an eigenvector for
the eigenvalue 1,v2
for 2, andv3
for 3, then what isA
(v1
+v2 v3
)?ANSWER:
A(v1 +v2
v3) = Av1 + A
v2 Av3 = 1
v1 + 2v2 3
v3
